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crib_sheet3 - Review for last part of Exam #3/Final...

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Review for last part of Exam #3/Final Statistics 211 1 Chapter 8: One Sample Hypothesis Tests One Sample test for mean μ σ is known: Z = ¯ x - μ 0 σ/ n Alternative hypothesis Rejection region for level α test H a : μ > μ 0 Z Z α H a : μ < μ 0 Z ≤ - Z α H a : μ ± = μ 0 Z Z α/ 2 or Z ≤ - Z α/ 2 Type II Error Formulas for β ( μ ± ) Alternative hypothesis Type II error for size α test H a : μ > μ 0 Φ z α + μ 0 - μ ± σ/ n · H a : μ < μ 0 1 - Φ - z α + μ 0 - μ ± σ/ n · H a : μ ± = μ 0 Φ z α/ 2 + μ 0 - μ ± σ/ n · - Φ - z α/ 2 + μ 0 - μ ± σ/ n · Sample sizes and Type II errors: Alternative hypothesis Sample size H a : μ > μ 0 h σ ( z α + z β ) μ - μ ± i 2 H a : μ < μ 0 h σ ( z α + z β ) μ - μ ± i 2 H a : μ ± = μ 0 h σ ( z α/ 2 + z β ) μ - μ ± i 2 One Sample Test for Mean μ , Large Sample (CLT): Z = ¯ x - μ 0 s/ n Rejection region is the same as one sample Z -test above. One sample test for mean μ σ is unknown: t = ¯ x - μ 0 s/ n Alternative hypothesis Rejection region for level α test H a : μ > μ 0 t t α,n - 1 H a : μ < μ 0 t ≤ - t α,n - 1 H a : μ ± = μ 0 t t α/ 2 ,n - 1 or t ≤ - t α/ 2 ,n - 1 1

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Large sample tests concerning a population proportion p : Z = ˆ θ - θ 0 σ ˆ θ = ˆ p - p 0 p p 0 q 0 /n Alternative hypothesis Rejection region for level α test H a : p > p 0 Z Z α H a : p < p 0 Z ≤ - Z α H a : p ± = p 0 Z Z α/ 2 or Z ≤ - Z α/ 2 Note that np 0 5 and nq 0 5 for this test to hold. Type II error formulas for β ( p ± ): Alternative hypothesis Type II error for size α test H a : p > p 0 Φ ± p 0 - p ± + Z α p 0 q 0 /n p ± q ± /n H a : p < p 0 1 - Φ ± p 0 - p ± - Z α p 0 q 0 /n p ± q ± /n H a : p ± = p 0 Φ ± p 0 - p ± + Z α/ 2 p 0 q 0 /n p ± q ± /n - Φ ± p 0 - p ± - Z α/ 2 p 0 q 0 /n p ± q ± /n Sample sizes and Type II errors: Alternative hypothesis Sample size
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This note was uploaded on 03/29/2008 for the course STAT 211 taught by Professor Parzen during the Spring '07 term at Texas A&M.

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crib_sheet3 - Review for last part of Exam #3/Final...

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